Given an arithmetic sequence we'll sometimes need to calculate the sum of its first \(n\) terms.
For example, given the arithmetic sequence whose first few terms are:
\[3,7,11,15,19,23, \dots \]
we may need to calculate the sum of its first \(100\) terms.
We could do this by adding one term to the next up to the \(100^{\text{th}}\) term but that would take time.
Instead we use one of two formula.
We have two formula for the sum of the first \(n\) terms of an arithmetic sequence, we learn both in this section.
Given an arithmetic sequence, we can calculathe the sum of its first \(n\) terms, which we write \(S_n\), using the formula:
\[S_n = \frac{n}{2} \begin{pmatrix} u_1 + u_n \end{pmatrix}\]
Where \(u_1\) is the first term of the sequence and \(u_n\) is the \(n^{\text{th}}\) term.
So, for example, \(S_{10}\) refers to the sum of the first \(10\) terms, \(S_{250}\) refers to the sum of the first \(250\) terms, ... .
The method for using this formula is illustrated in the tutorial below.
In the following tutorial we review the method for findng the formula for the \(n^{\text{th}}\) term of a linear sequence. Watch it now.
Given an arithmetic sequence, we can calculate the sum of its first \(n\) terms, \(S_n\), using the formula:
\[S_n = \frac{n}{2} \begin{pmatrix} 2.u_1 + (n-1)d \end{pmatrix}\]
Where \(u_1\) is the first term of the sequence and \(d\) is its common difference.
Note: exam questions frequently involve this second formula.
We now learn how to solve some tyical exam-type questions involving the sum of the first \(n\) terms of an arithmetic sequence.